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If the solution curve y = y(x) of the differential equation y2dx + (x2 - xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y = √3 x at the point (α, √3α), then value of loge(√3α) is equal to
- a)π/3
- b)π/2
- c)π/12
- d)π/6
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
If the solution curve y = y(x) of the differential equation y2dx + (x2...
Let's first rewrite the given differential equation:
y^2dx - (x^2 - xy - y^2)dy = 0
To find the solution curve, we can solve this differential equation. Let's separate the variables and integrate:
∫(y^2dx) - ∫(x^2 - xy - y^2)dy = ∫0dx
∫y^2dx - ∫(x^2 - xy - y^2)dy = 0
Let's evaluate each integral:
∫y^2dx = xy^2 + C1
∫(x^2 - xy - y^2)dy = x^2y - (1/2)xy^2 - (1/3)y^3 + C2
Substituting these results back into the original equation, we have:
xy^2 + C1 - (x^2y - (1/2)xy^2 - (1/3)y^3 + C2) = 0
Rearranging the terms, we get:
xy^2 + (x^2y - (1/2)xy^2 - (1/3)y^3) - C1 + C2 = 0
Combining like terms, we have:
(x^2 + 1/2xy - 1/3y^2)y - C1 + C2 = 0
We can rewrite this equation as:
(x^2 + 1/2xy - 1/3y^2)y = C3
where C3 = C1 - C2.
Now, let's find the value of C3 using the given point (1, 1):
(1^2 + 1/2(1)(1) - 1/3(1)^2)(1) = C3
(1 + 1/2 - 1/3)(1) = C3
(4/3)(1) = C3
C3 = 4/3
Therefore, the equation of the solution curve that passes through the point (1, 1) is:
(x^2 + 1/2xy - 1/3y^2)y = 4/3
y^2dx - (x^2 - xy - y^2)dy = 0
To find the solution curve, we can solve this differential equation. Let's separate the variables and integrate:
∫(y^2dx) - ∫(x^2 - xy - y^2)dy = ∫0dx
∫y^2dx - ∫(x^2 - xy - y^2)dy = 0
Let's evaluate each integral:
∫y^2dx = xy^2 + C1
∫(x^2 - xy - y^2)dy = x^2y - (1/2)xy^2 - (1/3)y^3 + C2
Substituting these results back into the original equation, we have:
xy^2 + C1 - (x^2y - (1/2)xy^2 - (1/3)y^3 + C2) = 0
Rearranging the terms, we get:
xy^2 + (x^2y - (1/2)xy^2 - (1/3)y^3) - C1 + C2 = 0
Combining like terms, we have:
(x^2 + 1/2xy - 1/3y^2)y - C1 + C2 = 0
We can rewrite this equation as:
(x^2 + 1/2xy - 1/3y^2)y = C3
where C3 = C1 - C2.
Now, let's find the value of C3 using the given point (1, 1):
(1^2 + 1/2(1)(1) - 1/3(1)^2)(1) = C3
(1 + 1/2 - 1/3)(1) = C3
(4/3)(1) = C3
C3 = 4/3
Therefore, the equation of the solution curve that passes through the point (1, 1) is:
(x^2 + 1/2xy - 1/3y^2)y = 4/3
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Community Answer
If the solution curve y = y(x) of the differential equation y2dx + (x2...
y2dx - xy dy = -(x2 + y2)dy
y(y dx - x dy) = - (x2 + y2)dy
- y (x dy - y dx) = - (x2 + y2)dy
y(y dx - x dy) = - (x2 + y2)dy
- y (x dy - y dx) = - (x2 + y2)dy
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If the solution curve y = y(x) of the differential equation y2dx + (x2 - xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y =√3x at the point (α, √3α), then value of loge(√3α) is equal toa)π/3b)π/2c)π/12d)π/6Correct answer is option 'C'. Can you explain this answer?
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If the solution curve y = y(x) of the differential equation y2dx + (x2 - xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y =√3x at the point (α, √3α), then value of loge(√3α) is equal toa)π/3b)π/2c)π/12d)π/6Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the solution curve y = y(x) of the differential equation y2dx + (x2 - xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y =√3x at the point (α, √3α), then value of loge(√3α) is equal toa)π/3b)π/2c)π/12d)π/6Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the solution curve y = y(x) of the differential equation y2dx + (x2 - xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y =√3x at the point (α, √3α), then value of loge(√3α) is equal toa)π/3b)π/2c)π/12d)π/6Correct answer is option 'C'. Can you explain this answer?.
If the solution curve y = y(x) of the differential equation y2dx + (x2 - xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y =√3x at the point (α, √3α), then value of loge(√3α) is equal toa)π/3b)π/2c)π/12d)π/6Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the solution curve y = y(x) of the differential equation y2dx + (x2 - xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y =√3x at the point (α, √3α), then value of loge(√3α) is equal toa)π/3b)π/2c)π/12d)π/6Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the solution curve y = y(x) of the differential equation y2dx + (x2 - xy + y2)dy = 0, which passes through the point (1, 1) and intersects the line y =√3x at the point (α, √3α), then value of loge(√3α) is equal toa)π/3b)π/2c)π/12d)π/6Correct answer is option 'C'. Can you explain this answer?.
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